Theorem exists1 1862

Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 3498.

Assertion

Ref	Expression
exists1	|- (E!x x = x <-> A.x x = y)

Distinct variable group:   x,y

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		df-eu 1775	. 2 |- (E!x x = x <-> E.yA.x(x = x <-> x = y))
2		equid 1484	. . . . . 6 |- x = x
3	2	tbt 788	. . . . 5 |- (x = y <-> (x = y <-> x = x))
4		bicom 579	. . . . 5 |- ((x = y <-> x = x) <-> (x = x <-> x = y))
5	3, 4	bitri 190	. . . 4 |- (x = y <-> (x = x <-> x = y))
6	5	albii 1346	. . 3 |- (A.x x = y <-> A.x(x = x <-> x = y))
7	6	exbii 1398	. 2 |- (E.yA.x x = y <-> E.yA.x(x = x <-> x = y))
8		hbae 1505	. . 3 |- (A.x x = y -> A.yA.x x = y)
9	8	19.9 1383	. 2 |- (E.yA.x x = y <-> A.x x = y)
10	1, 7, 9	3bitr2i 196	1 |- (E!x x = x <-> A.x x = y)

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298  E.wex 1326  E!weu 1771

This theorem is referenced by:  exists2 1863  exists2OLD 1864

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-eu 1775

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